Question

18. In a study of memory recall, 5 people were given 10 minutes to memorize a list of 20 words. Each was asked to list as many of the words as he or she could remember both 1 hour and 24 hours later. Does the data below suggest that the mean number of words recalled after 1 hour exceeds the mean recall after 24 hours? Assume we want to use a 0.05 significance level to test the claim.

Number of worlds recalled

Subject 1 hour later 24 hours later

1 14 12

2   18 14

3 11 8

4 13 12

5 12 12

(a) What is the appropriate hypothesis test to use for this analysis? Please identify and explain why it is appropriate.

(b) Let μ1 = mean words recalled after 1 hour. Let μ2 = mean words recalled after 24 hours. Which of the following statements correctly defines the null hypothesis?

(i) μ1 - μ2 > 0 (μd > 0)

(ii) μ1 - μ2 = 0 (μd = 0)

(iii) μ1 - μ2 < 0 (μd < 0)

(c) Let μ1 = mean words recalled after 1 hour. Let μ2 = mean words recalled after 24 hours. Which of the following statements correctly defines the alternative hypothesis?

(i) μ1 - μ2 > 0 (μd > 0)

(ii) μ1 - μ2 = 0 (μd = 0)

(iii) μ1 - μ2 < 0 (μd < 0)

(d) Determine the test statistic. Round your answer to three decimal places. Describe method used for obtaining the test statistic.

(e) Determine the p-value. Round your answer to three decimal places. Describe method used for obtaining the p-value.

(f) Compare p-value and significance level α. What decision should be made regarding the null hypothesis (e.g., reject or fail to reject) and why?

(g) What do the results of this study tell us about the mean number of words recalled after 1 hours and after 24 hours? Justify your conclusion.

Answer 1

Solution:

(a) What is the appropriate hypothesis test to use for this analysis? Please identify and explain why it is appropriate.

Answer: The appropriate hypothesis test to use for this analysis is paired t test, because given two samples are related samples.

(b) Let μ1 = mean words recalled after 1 hour. Let μ2 = mean words recalled after 24 hours. Which of the following statements correctly defines the null hypothesis?

Answer: The correct null hypothesis is given as below:

(ii) μ1 - μ2 = 0 (μd = 0)

(c) Let μ1 = mean words recalled after 1 hour. Let μ2 = mean words recalled after 24 hours. Which of the following statements correctly defines the alternative hypothesis?

Answer: The correct alternative hypothesis is given as below:

(i) μ1 - μ2 > 0 (μd > 0)

(d) Determine the test statistic. Round your answer to three decimal places. Describe method used for obtaining the test statistic.

Test statistic for paired t test is given as below:

t = (Dbar - µ_d)/[S_d/sqrt(n)]

From given data, we have

Dbar = 2

S_d = 1.5811

n = 5

df = n – 1 = 4

α = 0.05

Critical value = 2.1318

(by using t-table)

t = (Dbar - µ_d)/[S_d/sqrt(n)]

t = (2 - 0)/[ 1.5811/sqrt(5)]

t = 2/0.7071

t = 2.8284

Test statistic = t = 2.828

(e) Determine the p-value. Round your answer to three decimal places. Describe method used for obtaining the p-value.

We have test statistic t = 2.8284

df = n – 1 = 4

So, by using above two values in the t-table, the corresponding p-value is given as below:

P-value = 0.024

f) Compare p-value and significance level α. What decision should be made regarding the null hypothesis (e.g., reject or fail to reject) and why?

P-value = 0.024

α = 0.05

P-value < α = 0.05

So, we reject the null hypothesis

(g) What do the results of this study tell us about the mean number of words recalled after 1 hours and after 24 hours? Justify your conclusion.

There is sufficient evidence to conclude that that the mean number of words recalled after 1 hour exceeds the mean recall after 24 hours.